The question here is - can we build a cell whose only function is to self-replicate? No DNA error corrections (it might be interesting to see how fast the replication falls apart if we allowed this.) No unnecessary machinery outside of the ability to absorb/create new material to grow the cell for division, and to support the division process itself.
I believe we know the genes responsible for some or all of these processes already. Because I have no clue how free-floating DNA can create a cell-wall around it - a mystery of first life methinks, I think this can be achieved by inserting the code into some simple existing simple Prokaryotic cell. As the cells divide, all the old functionality and internal structures go away.
From these simple artificial cell forms, I think we can start to inject functionality into its DNA. This should allow us to establish a baseline to determine how (if) DNA code influences timing, the input/output functions in the form of concentrations of proteins and amino acids. I believe this had already been done at some higher level.